Sign-Changing and Multiple Solutions of Impulsive Boundary Value Problems Via Critical Point Methods

In this paper, we study the second-order impulsive boundary value problem $$\left\{\begin{array}{ll} -Lu=f(x, u), \, \, x\in [0, 1]\backslash\{x_{1}, x_{2}, \cdots, x_{l}\}, \\ -{\Delta} (p(x_{i}) u'(x_{i}))=I_{i}(u(x_{i})), \quad i=1, 2, \cdots, l, \\ R_{1}(u)=0, \, \, \, R_{2}(u)=0, \end{array}\right.$$ where Lu = (p(x)u′)′ ? q(x)u is a Sturm-Liouville operator, R ₁(u) = αu′(0) ? βu(0) and R ₂(u) = γu′(1) + σu(1). The existence of sign-changing and multiple solutions is obtained. The technical approach is based on minimax methods and invariant sets of descending flow.     

Second-Order Viability Problems with Perturbation in Hilbert Spaces

In this paper, we study the existence of viable solutions to the differential inclusion

$ \ddot{x}(t) \in f\left( {t,x(t),\dot{x}(t)} \right) + F\left( {x(t),\dot{x}(t)} \right), $

where f is a Carathéodory single-valued map and F is an upper semi-continuous multifunction with compact values contained in the Clarke subdifferential ? _c V of an uniformly regular function V.

     

Analytic Solutions of Systems

In this paper, functional series solutions of the nonlinear analytic system for the unknown state variable x(t), and functional series solutions of the analytic infinite-dimension

Controllability of the Semilinear Beam Equation

In this paper, we prove the approximate controllability of the following semilinear beam equation: $$ \left\{ \begin{array}{lll} \displaystyle{\partial^{2} y(t,x) \over \partial t^{2}} & = & 2\beta\Delta\displaystyle\frac{\partial y(t,x)}{\partial t}- \Delta^{2}y(t,x)+ u(t,x) + f(t,y,y_{t},u),\; \mbox{in}\; (0,\tau)\times\Omega, \\ y(t,x) & = & \Delta y(t,x)= 0 , \ \ \mbox{on}\; (0,\tau)\times\partial\Omega, \\ y(0,x) & = & y_{0}(x), \ \ y_{t}(x)=v_{0}(x), x \in \Omega, \end{array} \right. $$ in the states space $Z_{1}=D(\Delta)\times L^{2}(\Omega)$ with the graph norm, where β?>?1, Ω is a sufficiently regular bounded domain in IR ^N , the distributed control u belongs to L ²([0,τ];U) (U?=?L ²(Ω)), and the nonlinear function $f:[0,\tau]\times I\!\!R\times I\!\!R\times I\!\!R\longrightarrow I\!\!R$ is smooth enough and there are a,c?∈?IR such that $a<\lambda_{1}^{2}$ and $$ \displaystyle\sup\limits_{(t,y,v,u)\in Q_{\tau}}\mid f(t,y,v,u) - ay -cu\mid<\infty, $$ where Q _τ ?=?[0,τ]×IR×IR×IR. We prove that for all τ?>?0, this system is approximately controllable on [0,τ].     

Explicit solutions of the {\mathfrak{a}_1} -type lie-Scheffers system and a general Riccati equation

For a general differential system $ \dot{x}(t) = \sum\nolimits_{d = 1}^3 {u_d } (t){X_d} $ , where X _d generates the simple Lie algebra of type $ {\mathfrak{a}_1} $ , we compute the explicit solution in terms of iterated integrals of products of u _d ’s. As a byproduct we obtain the solution of a general Riccati equation by infinite quadratures.     

Nonoscillation and Exponential Stability of Delay Differential Equations with Oscillating Coefficients

We prove that under some additional conditions, the nonoscillation of the scalar delay differential equation

Multiple positive solutions of singular dirichlet second-order boundary-value problems with derivative dependence

The existence of multiple positive solutions for the singular Dirichlet boundary-value problem

Limit Cycles for a Class of Three-Dimensional Polynomial Differential Systems

Perturbing the system inside the family of polynomial differential systems of degree n in , we obtain at most n ² limit cycles using the first-order averaging theory. Moreover, there exist such perturbed systems having at least n ² limit cycles.      

Special representations of ℤ n -actions

In this paper we generalize the following statement (Alpern's theorem). Given a relatively prime set $$\{ h_i \} \subset \mathbb{N},i = 1,...,N \leqslant \omega ,$$ , and a probability distribution {α _i }, for any antiperiodicT there is a representation $X = \coprod\nolimits_{i = 1}^N {\left( {\coprod\nolimits_{j = 0}^{h_i - 1} {T^j B_i } } \right)}$ , where μ(B _i )=α _i /h _i . Our main result is the similar statement for free ? ⁿ -actions. Both theorems are generalizations of the well-known Rokhlin-Halmos lemma.     

Exact boundary zero controllability of three-dimensional Navier-Stokes equations

In a bounded three-dimensional domain Ω a solenoidal initial vector fieldv ₀(x)∈H ³ (Ω) is given. We construct a vector fieldz(t, x) defined on the lateral surface [0,T]×?Ω of the cylinder [0,T]×Ω which possesses the following property: the solutionv(t, x) of the boundary value problem for the Navier-Stokes equation with the initial valuev ₀(x) and the boundary Dirichlet conditionz(t, x) satisfies the relationv(T, x)≡0 at the instantT. Moreover, $$\parallel v(t, \cdot )\parallel _{H^3 (\Omega )} \leqslant c\exp \left( { - k/(T - t)^2 } \right) as t \to T$$ , wherec>0,k>0 are certain constants.     

Homogeneous tangent vectors and high-order necessary conditions for optimal controls

An extension of the classical Pontryagin maximum principle for Mayer problems without terminal constraints, subject to affine control systems , is proved. In connection with a suitable dilation on the state space ℝ ⁿ , we introduce a class of “homogeneous tangent vectors” which provide a nonlinear, high-order approximation of the attainable set in the case where the usual linear approximation proves to be inadequate. By studying control variations which generate homogeneous tangent vectors, we derive new necessary conditions for optimality that are particularly effective for basically nonlinear optimal control problems where other high-order tests provide no conclusive information. This research was partially supported by Istituto Nazionale di Alta Matematica “F. Severi.”     

Some Properties of the Value Function and Its Level Sets for Affine Control Systems with Quadratic Cost

Let be fixed. We consider the optimal control problem for analytic affine systems: , with a cost of the form: . For this kind of systems we prove that if there are no minimizing abnormal extremals then the value function S is subanalytic. Second, we prove that if there exists an abnormal minimizer of corank 1, then the set of endpoints of minimizers at cost fixed is tangent to a given hyperplane. We illustrate this situation in sub-Riemannian geometry.     

Optimal Control on the Heisenberg Group

Let H denote either the Heisenberg group , or the Cartesian product of n copies of the three-dimensional Heisenberg group . Let {X ₁, Y ₁, ...;, X _n, Y _n} be an independent set of left-invariant vector fields on H. In this paper, we study the left-invariant optimal control problem on H with the dynamics the cost functional with arbitrary positive parameters ₁, ...;, _n , and admissible controls taken from the set of measurable functions The above control system is encoded either in the kernel of a contact 1-form (for ), or in the kernel of a Pfaffian system (for ). In both cases, the action of the semi-direct product of the torus T ⁿ with H describe the symmetries of the problem.The Pontryagin maximum principle provides optimal controls; extremal trajectories are solutions to the Hamiltonian system associated with the problem. Abnormal extremals (which do not depend on the cost functional) yield solutions that are geometrically irrelevant.An explicit integration of the extremal equations provides a tool for studying some aspects of the sub-Riemannian structure defined on H by means of the above optimal control problem.     

Multifractal Analysis of Ergodic Averages: A Generalization of Eggleston's Theorem

Let V be a finite set, S be an infinite countable commutative semigroup, { _s , s S} be the semigroup of translations in the function space X = V ^S , A = {A _n } be a sequence of finite sets in S, f be a continuous function on X with values in a separable real Banach space B, and let B. We introduce in X a scale metric generating the product topology. Under some assumptions on f and A, we evaluate the Hausdorff dimension of the set X _f,,Adefined by the following formula:

Output-Based Stabilization of Timoshenko Beam with the Boundary Control and Input Distributed Delay

In this paper, we consider the output-feedback exponential stabilization of Timoshenko beam with the boundary control and input distributed delay. Suppose that the outputs of controllers are of the forms \(\alpha _{1}u_{1}(t)+\beta _{1}u_{1}(t-\tau )+{\int }_{-\tau }^{0}g_{1}(\eta )u_{1} (t+\eta )d\eta \) and \(\alpha _{2}u_{2}(t)+\beta _{2}u_{2}(t-\tau ) +{\int }_{-\tau }^{0}g_{2}(\eta )u_{2}(t+\eta )d\eta \) respectively, where u ₁(t) and u ₂(t) are the inputs of controllers. Using the tricks of the Luenberger observer and partial state predictor, we translate the system with delay into a system without delay. And then, we design the feedback controls to stabilize the system without delay. Finally, we prove that under the choice of such controls, the original system also is stabilized exponentially.     

Study on Existence of Solutions for p-Kirchhoff Elliptic Equation in ℝN with Vanishing Potential

In this paper, we study the existence of positive solutions to p?Kirchhoff elliptic problem \(\begin{array}{@{}rcl@{}} \left\{\begin{array}{lllllll} &\left(a+\mu\left({\int}_{\mathbb{R}^{N}}\!(|\nabla u|^{p}+V(x)|u|^{p})dx\right)^{\tau}\right)\left(-{\Delta}_{p}u+V(x)|u|^{p-2}u\right)=f(x,u), \quad \text{in}\; \mathbb{R}^{N}, \\ &u(x)>0, \;\;\text{in}\;\; \mathbb{R}^{N},\;\; u\in \mathcal{D}^{1,p}(\mathbb{R}^{N}), \end{array}\right.\!\!\!\! \\ \end{array} \) ?????(0.1) where a, μ > 0, τ > 0, and f(x, u) = h ₁(x)|u| ^m?2 u + λ h ₂(x)|u| ^r?2 u with the parameter λ ∈ ?, 1 < p < N, 1 < r < m < \(p^{*}=\frac {pN}{N-p}\) , and the functions h ₁ (x), h ₂(x) ∈ C(?^N) satisfy some conditions. The potential V(x) > 0 is continuous in ? ^N and V(x)→0 as |x|→+∞. The nontrivial solution forb Eq. (1.1) will be obtained by the Nehari manifold and fibering maps methods and Mountain Pass Theorem.     

Existence of Weak Solutions for Generalized Quasilinear Schrödinger Equations

This paper shows the existence of nontrivial weak solutions for the generalized quasilinear Schrödinger equations

$$ -div(g^{p}(u)|\nabla u|^{p-2}\nabla u)+g^{p-1}(u)g^{\prime}(u)|\nabla u|^{p}+ V(x)|u|^{p-2}u=h(u),\,\, x\in \mathbb{R}^{N}, $$

where N ≥ 3, \(g(s): \mathbb {R}\rightarrow \mathbb {R}^{+}\) is C ¹ nondecreasing function with respect to |s|, V is a positive potential bounded away from zero and h(u) is a nonlinear term of subcritical type. By introducing a variable replacement and using minimax methods, we show the existence of a nontrivial solution in \(C^{\alpha }_{loc}(\mathbb {R}^{N})\).

     

Periodic Solutions for a Class of Second Order Delay Systems

The main purpose of this paper is to study the existence and multiplicity of periodic solutions for the following non-autonomous second order delay systems $$ -\frac{{{d^2}u}}{{d{t^2}}}+Au(t)=f\left( {t,u\left( {t-r} \right)} \right), $$ where A = (a _ij ) _n×n is a symmetric matrix, f ∈ C(R × R ⁿ , R ⁿ ) is r-periodic in the first variable, and r > 0 is a given constant. Some existence and multiplicity theorems of 2r-periodic solutions of such systems are obtained via the linking theorem of Benci and Rabinowitz.     

On High Dimensional Schrödinger Equation with Quasi-Periodic Potentials

In this paper, we consider the high dimensional Schrödinger equation \( -\frac {d^{2}y}{dt^{2}} + u(t)y= Ey, y\in \mathbb {R}^{n}, \) where u(t) is a real analytic quasi-periodic symmetric matrix, \(E= \text {diag}({\lambda _{1}^{2}}, \ldots , {\lambda _{n}^{2}})\) is a diagonal matrix with λ _j >0,j=1,…,n, being regarded as parameters, and prove that if the basic frequencies of u satisfy a Bruno-Rüssmann’s non-resonant condition, then for most of sufficiently large λ _j ,j=1,…,n, there exist n pairs of conjugate quasi-periodic solutions.